Radial structure, inflow and central mass of stationary radiative galaxy clusters

We analyse the radial structure of self-gravitating spheres consisting of multiple interpenetrating fluids, such as the X-ray emitting gas and the dark halo of a galaxy cluster. In these dipolytropic models, the adiabatic dark matter sits in equilibrium, while the gas develops a gradual, smooth, quasi-stationary cooling flow. Both affect and respond to the collective gravitational field. We find that all subsonic, radially continuous, steady solutions require a non-zero minimum central point mass. For Mpc-sized haloes with 7–10 effective degrees of freedom (F2), the minimum central mass is compatible with observations of supermassive black holes. Smaller gas mass influxes enable smaller central masses for wider ranges of F2. The halo comprises a sharp spike around the central mass, embedded within a core of nearly constant density (at 101–102.5 kpc scales), with outskirts that attenuate and naturally truncate at finite radius (several Mpc). The gas density resembles a broken power law in radius, but the temperature dips and peaks within the dark core. A finite minimum temperature occurs due to gravitational self-warming, without cold mass dropout nor needing regulatory heating. X-ray emission from the intracluster medium mimics a β-model plus bright compact nucleus. Near-sonic points in the gas flow are bottlenecks to the allowed steady solutions; the outermost are at kpc scales. These sites may preferentially develop cold mass dropout during strong perturbations off equilibrium. Within the sonic point, the profile of gas specific entropy is flatter than s∝r1/2, but this is a shallow ramp and not an isentropic core. When F2 is large, the inner halo spike is only marginally Jeans stable in the central parsec, suggesting that a large non-linear disturbance could trigger local dark collapse on to the central object

Topics in elliptic problems: from semilinear equations to shape optimization

In this paper, which corresponds to an updated version of the author's Habilitation lecture in Mathematics, we do an overview of several topics in elliptic problems. We review some old and new results regarding the Lane-Emden equation, both under Dirichlet and Neumann boundary conditions, then focus on sign-changing solutions for Lane-Emden systems. We also survey some results regarding fully nontrivial solutions to gradient elliptic systems with mixed cooperative and competitive interactions. We conclude by exhibiting results on optimal partition problems, with cost functions either related to Dirichlet eigenvalues or to the Yamabe equation. Several open problems are referred along the text.Comment: Review article focused on the author's own work(expanded version of his Habilitation lecture).Draws heavily from: arXiv:2305.02870,arXiv:2211.04839,arXiv:2209.02113, arXiv:2109.14753,arXiv:2106.03661,arXiv:2106.00579,arXiv:1908.11090,arXiv:1807.03082, arXiv:1706.08391, arXiv:1701.05005, arXiv:1508.01783,arXiv:1412.4336,arXiv:1409.5693,arXiv:1405.5549,arXiv:1403.6313,arXiv:1307.3981,arXiv:1201.520

On the general relativistic Thomas-Fermi theory of white dwarfs and neutron stars

We present a review of the multi-year effort in the formulation of a self-consistent theory for the description of white dwarfs and neutron stars based on the general relativistic extension of the Thomas-Fermi model of the atom

Self-Similar Dynamics of a Relativistically Hot Gas

In the presence of self-gravity, we investigate the self-similar dynamics of a relativistically hot gas with or without shocks in astrophysical processes of stellar core collapse, formation of compact objects, and supernova remnants with central voids. The model system is taken to be spherically symmetric and the conservation of specific entropy along streamlines is adopted for a relativistic hot gas. In terms of equation of state, this leads to a polytropic index $\gamma=4/3$. The conventional polytropic gas of $P=\kappa\rho^\gamma$, where $P$ is the thermal pressure, $\rho$ is the mass density, $\gamma$ is the polytropic index, and $\kappa$ is a global constant, is included in our theoretical model framework. Two qualitatively different solution classes arise according to the values of a simple power-law scaling index $a$, each of which is analyzed separately and systematically. We obtain new asymptotic solutions that exist only for $\gamma=4/3$. Global and asymptotic solutions in various limits as well as eigensolutions across sonic critical lines are derived analytically and numerically with or without shocks. By specific entropy conservation along streamlines, we extend the analysis of Goldreich & Weber for a distribution of variable specific entropy with time $t$ and radius $r$ and discuss consequences in the context of a homologous core collapse prior to supernovae. As an alternative rebound shock model, we construct an Einstein-de Sitter explosion with shock connections with various outer flows including a static outer part of a singular polytropic sphere (SPS). Under the joint action of thermal pressure and self-gravity, we can also construct self-similar solutions with central spherical voids with sharp density variations along their edges.Comment: 21 pages, 15 figures, accepted for publication in MNRA

Approximate Universal Relations for Neutron Stars and Quark Stars

Neutron stars and quark stars are ideal laboratories to study fundamental physics at supra nuclear densities and strong gravitational fields. Astrophysical observables, however, depend strongly on the star's internal structure, which is currently unknown due to uncertainties in the equation of state. Universal relations, however, exist among certain stellar observables that do not depend sensitively on the star's internal structure. One such set of relations is between the star's moment of inertia ($I$), its tidal Love number (Love) and its quadrupole moment ($Q$), the so-called I-Love-Q relations. Similar relations hold among the star's multipole moments, which resemble the well-known black hole no-hair theorems. Universal relations break degeneracies among astrophysical observables, leading to a variety of applications: (i) X-ray measurements of the nuclear matter equation of state, (ii) gravitational wave measurements of the intrinsic spin of inspiraling compact objects, and (iii) gravitational and astrophysical tests of General Relativity that are independent of the equation of state. We here review how the universal relations come about and all the applications that have been devised to date.Comment: 89 pages, 38 figures; review article submitted to Physics Report

An Accurate Solution of the Self-Similar Orbit-Averaged Fokker-Planck Equation for Core-Collapsing Isotropic Globular Clusters: Properties and Application

Hundreds of dense star clusters exist in almost all galaxies. Each cluster is composed of approximately ten thousand through ten million stars. The stars orbit in the clusters due to the clusters\u27 self-gravity. Standard stellar dynamics expects that the clusters behave like collisionless self-gravitating systems on short time scales (~ million years) and the stars travel in smooth continuous orbits. Such clusters temporally settle to dynamically stable states or quasi-stationary states (QSS). Two fundamental QSS models are the isothermal- and polytropic- spheres since they have similar structures to the actual core (central part) and halo (outskirt) of the clusters. The two QSS models are mathematically modeled by the Lane-Emden equations. On long time scales (~ billion years), the clusters experience a relaxation effect (Fokker-Planck process). This is due to the finiteness of total star number in the clusters that causes stars to deviate from their smooth orbits. This relaxation process forms a highly-dense relaxed core and sparse-collisionless halo in a self-similar fashion. The corresponding mathematical model is called the self-similar Orbit-Averaged Fokker-Planck (ss-OAFP) equation. However, any existing numerical works have never satisfactorily solved the ss-OAFP equation last decades after it was proposed. This is since the works rely on finite difference (FD) methods and their accuracies were not enough to cover the large gap in the density of the ss-OAFP model. To overcome this numerical problem, we employ a Chebyshev pseudo-spectral method. Spectral methods are known to be accurate and efficient scheme compared with FD methods. The present work proposes a new method by combining the Chebyshev spectral method with an inverse mapping of variables. Our new method provides accurate numerical solutions of the Lane-Emden equations with large density gaps on MATLAB software. The maximum density ratio of the core to halo can reach the possible numerical (graphical) limit of MATLAB. The same method provides four significant figures of a spectral solution to the ss-OAFP equation. This spectral solution infers that existing solutions have at most one significant figure. Also, our numerical results provide three new findings. (i) We report new kinds of the end-point singularities for the Chebyshev expansion of the Lane-Emden- and ss-OAFP equations. (ii) Based on the spectral solution, we discuss the thermodynamic aspects of the ss-OAFP model and detail the cause of the negative heat capacity of the system. We suggest that to hold a \u27negative\u27 heat capacity over relaxation time scales stars need to be not only in a deep potential well but also in a non-equilibrium state with the flow of heat and stars. (iii) We propose an energy-truncated ss-OAFP model that can fit the observed structural profiles of at least half of Milky Way globular clusters. The model can apply to not only normal clusters but also post collapsed-core clusters with resolved (observable) cores; those clusters can not generally be fitted by a single model. The new model is phenomenological in the sense that the energy-truncation is based on polytropic models while the truncation suggests that low-concentration globular clusters are possibly polytropic clusters